Abstract
We study the elliptic genera of two-dimensional orbifold CFTs, where the orbifolding procedure introduces du Val surface singularities on the target space. The \( \mathcal{N}=4 \) characterdecompositionsoftheellipticgenuscontributionsfromthetwistedsectors at the singularities obey a consistent scaling property, and contain information about the arrangement of exceptional rational curves in the resolution. We also discuss how these twisted sector elliptic genera are related to twining genera and Hodge elliptic genera for sigma models with K3 target space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
J. McKay, Graphs, singularities and finite groups, Proc. Symp. Pure Math. 37 (1980) 183.
M. Reid, McKay correspondence, alg-geom/9702016 [INSPIRE].
Y.-H. He and J.S. Song, Of McKay correspondence, nonlinear σ-model and conformal field theory, Adv. Theor. Math. Phys. 4 (2000) 747 [hep-th/9903056] [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu twining characters for K3, JHEP 09 (2010) 058 [arXiv:1006.0221] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral moonshine, Commun. Num. Theor. Phys. 08 (2014) 101 [arXiv:1204.2779] [INSPIRE].
E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].
D. Anselmi, M. Billó, P. Fré, L. Girardello and A. Zaffaroni, ALE manifolds and conformal field theories, Int. J. Mod. Phys. A 9 (1994) 3007 [hep-th/9304135] [INSPIRE].
T. Eguchi, Y. Sugawara and A. Taormina, Liouville field, modular forms and elliptic genera, JHEP 03 (2007) 119 [hep-th/0611338] [INSPIRE].
J.A. Harvey, S. Lee and S. Murthy, Elliptic genera of ALE and ALF manifolds from gauged linear σ-models, JHEP 02 (2015) 110 [arXiv:1406.6342] [INSPIRE].
T. Eguchi and A. Taormina, Unitary representations of N = 4 superconformal algebra, Phys. Lett. B 196 (1987) 75 [INSPIRE].
T. Eguchi and A. Taormina, Character formulas for the N = 4 superconformal algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].
S. Kachru and A. Tripathy, The Hodge-elliptic genus, spinning BPS states and black holes, arXiv:1609.02158 [INSPIRE].
S. Kachru and A. Tripathy, BPS jumping loci and special cycles, arXiv:1703.00455 [INSPIRE].
N. Benjamin, A refined count of BPS states in the D1/D5 system, JHEP 06 (2017) 028 [arXiv:1610.07607] [INSPIRE].
L. Borisov and A. Libgober, Elliptic genera of singular varieties, Duke Math. J. 116 (2003) 319 [math/0007108] [INSPIRE].
L. Borisov and A. Libgober, Elliptic genera of singular varieties, orbifold elliptic genus and chiral de Rham complex, math/0007126.
L. Borisov and A. Libgober, McKay correspondence for elliptic genera, Ann. Math. 161 (2005) 1521 [math/0206241].
R. Waelder, Equivariant elliptic genera and local McKay correspondence, Asian J. Math. 12 (2008) 251 [math/0701336].
G. Xiao, Galois covers between K3 surfaces, Ann. Inst. Fourier 46 (1996) 73.
P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, in Mirror symmetry II, B. Greene and S.-T. Yau eds., International Press, (1997), pg. 703 [hep-th/9404151] [INSPIRE].
W. Nahm and K. Wendland, A hiker’s guide to K3: aspects of N = (4,4) superconformal field theory with central charge c = 6, Commun. Math. Phys. 216 (2001) 85 [hep-th/9912067] [INSPIRE].
K. Wendland, A family of SCFTs hosting all very attractive relatives of the (2)4 Gepner model, JHEP 03 (2006) 102 [hep-th/0512223] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K3 σ-models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
D. Huybrechts, On derived categories of K3 surfaces, symplectic automorphisms and the Conway group, Adv. Stud. Pure. Math. 69 (2016) 387 [arXiv:1309.6528] [INSPIRE].
K. Wendland, Consistency of orbifold conformal field theories on K3, Adv. Theor. Math. Phys. 5 (2002) 429 [hep-th/0010281] [INSPIRE].
P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329 [hep-th/9507012] [INSPIRE].
V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, Alg. Analiz 11 (1999) 100 [St. Petersburg Math. J. 11 (2000) 781] [math/9906190] [INSPIRE].
T. Creutzig and G. Höhn, Mathieu moonshine and the geometry of K3 surfaces, Commun. Num. Theor. Phys. 08 (2014) 295 [arXiv:1309.2671] [INSPIRE].
A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing and mock modular forms, arXiv:1208.4074 [INSPIRE].
S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988) 183.
S. Kondo, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces, Duke Math. J. 92 (1998) 593.
M. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes I, Ann. Math. 86 (1967) 374.
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral moonshine and the Niemeier lattices, arXiv:1307.5793 [INSPIRE].
M.C.N. Cheng and S. Harrison, Umbral moonshine and K3 surfaces, Commun. Math. Phys. 339 (2015) 221 [arXiv:1406.0619] [INSPIRE].
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
C.V. Johnson and R.C. Myers, Aspects of type IIB theory on ALE spaces, Phys. Rev. D 55 (1997) 6382 [hep-th/9610140] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1704.02926
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wong, K. Quarter-BPS states in orbifold sigma models with ADE singularities. J. High Energ. Phys. 2017, 116 (2017). https://doi.org/10.1007/JHEP06(2017)116
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2017)116